Deutsch

Superpyth | Suprapyth | Music

---

Superpyth, a member of the Archytas clan, has 4/3 as a generator, and the Archytas comma 64/63 is tempered out, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for meantone and 12edo, with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). An interesting coincidence is that the plastic numberhas a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.

If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is "the opposite of" septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.

If intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98.
This temperament is called "supra", or "suprapyth" if you include 5 as well.

MOSes include 5, 7, 12, 17, and 22.

Superpyth

Commas: 64/63, 245/243

POTE generator: ~3/2 = 710.291

Map: [<1 0 -12 6|, <0 1 9 -2|]
Wedgie: <<1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323

11-limit

Commas: 64/63, 100/99, 245/243

POTE generator: ~3/2 = 710.175

Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
EDOs: 22, 27e, 49
Badness: 0.0250

13-limit

Commas: 64/63, 78/77, 91/90, 100/99

POTE generator: ~3/2 = 710.479

Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
EDOs: 22, 27e, 49, 76bcde
Badness: 0.0247

Suprapyth

Commas: 55/54, 64/63, 99/98

POTE generator: ~3/2 = 709.495

Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
EDOs: 5, 17, 22
Badness: 0.0328

Interval chains

Basic superpyth (2.3.7)

1146.61	437.29	927.97	218.64	709.32	0	490.68	981.36	272.03	762.71	53.39
27/14	9/7	12/7	9/8~8/7	3/2	1/1	4/3	7/4~16/9	7/6	14/9	28/27

Full 7-limit superpyth

613.20	1102.91	392.62	882.33	172.04	661.75	1151.46	441.16	930.87	220.58	710.29	0	489.71	979.42	269.13	758.84	48.54	538.25	1027.96	317.67	807.38	97.09	586.80
10/7	15/8	5/4	5/3	10/9		27/14	9/7	12/7	9/8~8/7	3/2	1/1	4/3	7/4~16/9	7/6	14/9	28/27		9/5	6/5	8/5	16/15	7/5

Supra (2.3.7.11)

857.54	150.35	643.15	1135.96	428.77	921.58	214.38	707.19	0	492.81	985.62	278.42	771.23	64.04	556.85	1049.65	342.46
18/11	12/11	16/11	27/14	14/11~9/7	12/7	9/8~8/7	3/2	1/1	4/3	7/4~16/9	7/6	14/9~11/7	33/32~28/27	11/8	11/6	11/9

Full 11-limit suprapyth

604.44	1094.94	385.45	875.96	166.46	656.97	1147.47	437.98	928.48	218.99	709.49	0	490.51	981.01	271.52	762.02	52.53	543.03	1033.54	324.04	814.55	105.06	595.56
10/7	15/8	5/4	18/11~5/3	12/11~10/9	16/11	27/14	14/11~9/7	12/7	9/8~8/7	3/2	1/1	4/3	7/4~16/9	7/6	14/9~11/7	33/32~28/27	11/8	9/5~11/6	6/5~11/9	8/5	16/15	7/5

MOSes

5-note (LsLss, proper)

See 2L 3s.

7-note (LLLsLLs, improper)

See 5L 2s. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.

12-note (LsLsLssLsLss, borderline improper)

See 5L 7s. The boundary of propriety is 17edo.

Music

12of22studyPentUp4thsMstr
12of22gamelan1b
12of22study3 (children's story)
12of22study7
By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.